< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)

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Download "< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)"

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1 < 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3) 6 y = g(x) x = 1 g( 1) = 2 ( 1) 3 = 2 ; g 0 ( 1) = 6 ( 1) 2 =6 ; ( 1; 2) 6 x =1 g(1) = 2 ; g 0 (1) = 6 ; (1; 2) 6 1

2 < 2 1 > f 0 (x) = lim h!0 f(x + h) f(x) h 2x 2 +4xh +2h 2 x h 2x 2 + x = lim h!0 h = lim(4x +2h 1) = 4x 1 h!0 = lim h!0 2(x + h) 2 (x + h) (2x 2 x) h = lim h!0 4xh +2h 2 h h 2

3 < 3 2 > (1) f(x) =x; f 0 (x) =lim h!0 x + h x h =1 (2) f(x) =x 2 ; f 0 (x) = lim h!0 (x + h) 2 x 2 h (3) f(x) =1; f (x) = lim =0 h!0 h =2x 3

4 < 4 > 1 (1) (a + b) 4 =(a + b)(a + b) 3 =(a + b)(a 3 +3a 2 b +3ab 2 + b 3 ) = 1 a a 3 b + 6 a 2 b ab b 4 ³ (2) (a + b) 5 =(a + b) 1 a a 3 b + 6 a 2 b ab b 4 = 1 a a 4 b + 10 a 3 b a 2 b ab b 5 2 (a + b) 0 =1 1 (a + b) 1 =1 a +1 b 1 1 (a + b) 2 =1 a 2 +2 ab +1 b (a + b) 3 =1 a 3 +3 a 2 b +3 ab 2 +1 b (a + b) 4 = 1 a a 3 b + 6 a 2 b ab b (a + b) 5 = 1 a a 4 b + 10 a 3 b a 2 b ab b (a + b) 6 = 1 a a 5 b + 15 a 4 b a 3 b a 2 b ab b 6 4

5 < 5 3 > 1 2 f 0 (x) = lim h!0 (x + h) 5 x 5 h x 5 +5x 4 h +10x 3 h 2 +10x 2 h 3 +5xh 4 + h 5 x 5 = lim h!0 h = lim 5x 4 +10x 3 h +10x 2 h 2 +5xh 3 + h 4 h!0 =5x 4 f 0 (x) = lim h!0 (x + h) 6 x 6 h x 6 +6x 5 h +15x 4 h 2 +20x 3 h 3 +15x 2 h 4 +6xh 5 + h 6 x 6 = lim h!0 h = lim 6x 5 +15x 4 h +20x 3 h 2 +15x 2 h 3 +6xh 4 + h 5 h!0 =6x 5 5

6 < 6 4 > 1 y x x 2 x 3 x 4 y 0 1 2x 3x 2 4x 3 (x) 0 = 1 (x 3 ) 0 = 3x 2 (x 4 ) 0 = 4x 3 2 (x n ) 0 = nx n 1 3 y 0 ah = lim h!0 h = a 6

7 < 7 5 > 1 (mx + k) 0 = m 2 (k) 0 =0 3 (kx 3 ) 0 =3kx 2 4 (kx n ) 0 = nkx n 1 5 fkf(x)g 0 = kf 0 (x) 7

8 < 8 6 > 1 (1) x =3x 2 (2) x 4 +2x 5 0 =4x 3 +10x 4 ³ 1 (3) 2 x2 x +2 0 = x 1 (4) x n x n+1 + k 0 = nx n 1 (n +1)x n 2 (1) f(x)+g(x) ª0 = f 0 (x)+g 0 (x) (2) f(x) g(x) ª0 = f 0 (x) g 0 (x) 8

9 < 9 > 1 y = m(x a)+b 2 y 0 =1 2x x =1 y 0 =1 2= 1 : y = 1(x 1) + 0 = x +1 3 y = f 0 (a)(x a)+b 9

10 < 10 1 > 1 (1) y = 2x 2 4x +5 y 0 = 4x 4= 4(x +1) x x< <x y y 00 % 7 & ( 1 ; 7) (2) y = 1 2 x2 +2 y 0 = x x x<0 0 0 <x y y 00 & 2 % (0 ; 2) 10

11 < 11 2 > 1 (1) y =2 3x + x 3 y 0 = 3+3x 2 =3(x 2 1) x 1 1 y y % 4 & 0 % x = 1 y =4 x = 1 y =0 (2) y =9x +3x 2 x 3 y 0 =9+6x 3x 2 = 3(x 2 2x 3) = 3(x 3)(x +1) x 1 3 y y & 5 % 27 & x = 3 y =27 x = 1 y = 5 11

12 < 12 1 > y =3x 4 +4x 3 12x x 5 2 y 0 =12x 3 +12x 2 24x =12x(x 2 + x 2) =12x(x +2)(x 1) x = 2 y =32 x = 1 y = 13 12

13 < 13 2 > y = x(2a 2x) 2 =4x(x 2 2ax + a 2 )=4x 3 8ax 2 +4a 2 x y 0 =12x 2 16ax +4a 2 =4(3x 2 4ax + a 2 )=4(3x a)(x a) 2a 2x >0 x 0 <x<a x = a 3 y =4x(a x)2 = 4 3 a(a a 3 )2 = 4 3 a µ 2 3 a 2 = a3 x = a a3 (cm 3 ) 13

14 < 14 > 1 x y y = jxj x = 0 y = x x<0 y = x 8 >< x (x = 0) y = jxj = >: x (x <0) 2 x y y = jx 2 4j 8 >< y = jx 2 4j = >: x 2 4 ( 2 5 x) x 2 +4 ( 2 <x< 2 ) x 2 4 (x 5 2 ) y= x 2 4 x x 14

15 < 15 > 1 (1) [5:98] = 5 (2) [ 3:01] = 4 (3) h 3 =1 (4) 2i h 11 5 i =

16 < 16 1 > (1) lim [x] = 1 lim x! 0 [x] =0 x!+0 (2) lim tan µ =+1 lim tan µ = 1 µ! ¼ 2 0 µ! ¼

17 < 17 2 > (1) jxj jxj lim = 1 (2) lim x! 0 x x!+0 x =1 17

18 < 18 1 > 1 `1 = r sin µ 2 `3 = r tan µ 3 `2 =2¼r µ 2¼ = rµ 4 r sin µ<rµ<rtan µ sin µ< µ <tan µ 5 sin µ < µ < sin µ cos µ 18

19 < 19 2 > 1 cos µ < sin µ µ < sin µ lim µ!+0 µ sin µ lim µ! 0 µ =1 sin( µ 1 ) sin(µ 1 ) sin(µ 1 ) = lim = lim = lim = 1 µ 1!+0 µ 1 µ 1!+0 µ 1 µ 1!+0 µ 1 19

20 < 20 3 > cos(x + µ) cos x cos x cos µ sin x sin µ cos x lim = lim µ!0 µ µ!0 µ ½ ¾ cos µ 1 = lim (cos x) (sin x) sin µ µ!0 µ µ = 1 0 (sin x) 1 = sin x 20

21 < 21 > 1 (cos x) 0 = sin x 2 (1) (3 cos x 2sinx) 0 = 3sinx 2cosx (2) (10 2x +sinx 5cosx) 0 = 2+cosx +5sinx 21

22 < 22 1 > f(x) g(x) 0 = f 0 (x) g(x)+f(x) g 0 (x) 22

23 < 23 2 > µ 0 1 = g0 (x) g(x) 2 g(x) 23

24 < 24 3 > 1 µ f(x) 0 = f 0 (x)g(x) f(x)g 0 (x) g(x) 2 g(x) 2 µ 0 1 = g0 (x) g(x) 2 g(x) 24

25 < 25 4 > 1 (1) (x 2 2x 1)(2x 2 + x +1) ª0 =(2x 2)(2x 2 + x +1)+(x 2 2x 1)(4x +1) =(4x 3 2x 2 2) + (4x 3 8x 2 + x 2 4x 2x 1) =8x 3 9x 2 6x 3 (2) µ 0 1 = 1+sinx x cos x (x cos x) 2 (3) µ 0 sin x = x (cos x) x sin x x 2 = x cos x sin x x 2 25

26 < 26 > 1 f(3) f(2) 3 2 = 44:1 19:6 3 2 =24:5 (m=s) 2 f 0 (2) = lim h!0 f(2 + h) f(2) h = lim h!0 4:9 4h +4:9h 2 h = lim h!0 4:9 (2 + h) 2 4:9 2 2 h =19:6 3 f 0 f(t + h) f(t) 4:9 (t + h) 2 4:9 t 2 (t) = lim =lim h!0 h h!0 h = lim h!0 4:9 2th +4:9h 2 h =4:9 2t =9:8t 26

27 < 27 > (1) y =2x 2 3x +4 (2) y =10 9:8t dy dx =4x 3 dy dt = 9:8 (3) ` =2¼r d` dr =2¼ (4) S = ¼r 2 (¼ ) ds dr =2¼r (5) V = 4 3 ¼r3 dv dr =4¼r2 27

28 < 28 > (x + x) 5 x 5 (1) lim x!0 x =(x 5 ) 0 =5x 4 cos(t + t) cos(t) (2) lim =(cost) 0 = sin t t!0 t tan(u + u) tan(u) (3) lim = (tan u) 0 = 1 u!0 u cos 2 u 28

29 < 29 > 1 (1) f(x) =x 2 1 ;g(x) =x +1 ; g f(x) =(x 2 1) + 1 = x 2 ; f g(x) =(x +1) 2 1=x 2 +2x (2) f(x) =2x; g(x) =cosx 1 ; g f(x) =cos(2x) 1 ; f g(x) =2(cosx 1) (3) f(x) =x 2 ;g(x) = p x ; g f(x) = p x 2 = x ; f g(x) =( p x) 2 = x (4) f(x) =2 x ;g(x) =log 2 x ; g f(x) =log 2 (2 x )=x ; f g(x) =2 log 2 x = x 2 (1) y =(x 2 3x +1) 3 (2) y =cos(2x 3) (3) y = p 1 x 2 (4) y =2 x2 1 ; f(x) =x 2 3x +1 ; f(x) =2x 3 ; f(x) =1 x 2 ; f(x) =x 2 1 ; g(x) =x 3 ; g(x) =cosx ; g(x) = p x ; g(x) =2 x 29

30 < 30 1 > dy dx = (cosu)0 (x 5 ) 0 = sin(u) 5x 4 = 5x 4 sin(x 5 ) 30

31 < 31 1 > 1 ( ) dy dx = dy du du dx 2 (1) y =(x 2 3x +1) 5 ; dy dx =5(2x 3)(x2 3x +1) 4 (2) y =cos(2x 3) ; dy dx = 2sin(2x 3) (3) y = p 1 x 2 ; dy dx = x p 1 x 2 31

32 < 32 1 > (1) µ (m+1) 1 1+ = (m +1) µ (m+1) m = m +1 µ m +1 m m+1 = µ 1+ 1 m m µ 1+ 1 m µ (2) lim 1+ 1 n µ (m+1) 1 = lim 1+ n! 1 n m!+1 (m +1) µ = lim 1+ 1 m m!+1 m = e 1=e µ 1+ 1 m 32

33 < 33 2 > 1 lim (1 + t) 1 t = t! 0 µ lim 1+ 1 x = e x! 1 x 2 (1) lim(1 + t) 1 t = e t!0 1 (2) lim t!0 t log e(1 + t) = lim log e (1 + t) 1 t =log e e =1 t!0 33

34 < 34 1 > f 0 1 (3) = lim h!0 h log 10 µ 3+h 3 µ h 3 = t 1 = lim t!0 3t log 1 10 (1 + t) = lim t!0 3 log 10 (1 + t) 1 t = 1 3 log 10 e f 0 (x)= 1 x log 10 e 34

35 < 35 2 > 1 ( ) (log a x) 0 = 1 x log a e 2 (1) log(e 2 )=2 (2) log µ 1 = 1 e (3) log 1 = 0 3 ( ) (log x) 0 = 1 x 35

36 < 36 3 > 1 (1) y =log(x 3 2x 1) dy dx = 1 x 3 2x 1 (x3 2x 1) 0 = 3x2 2 x 3 2x 1 (2) y = log(1 + cos x) dy dx = sin x 1+cosx (3) y =log(x sin x) dy dx = 1 cos x x sin x 2 ( ) ³ log f(x) 0 = f 0 (x) f(x) 36

37 < 37 > 1 log y = x log 3 y y 0 =log3 y = y 0 log 3 =3 x log 3 2 (a x ) 0 = a x log a 3 (e x ) 0 = e x log e = e x 37

38 < 38 x r > 1 log y = r log x y = r 1 y 0 x ) y0 = r 1 x y = r x 1 x r = rx r 1 (x r ) 0 = rx r 1 2 (1) ³ 3p 0 ³ 0 x 5 = x = 3 x = 3 3p x 2 (2) p x 0 = ³ 0 x = 1 2 x 1 2 = 2 p x (3) µ 0 1 ³ 0 px = x = 3 2 x 1 2 = 2x p x 38

39 < 39 log jxj > (1) y =logj tan xj ; dy dx = (tan x)0 tan x = 1 cos 2 x sin x cos x = 1 sin x cos x (2) y =logjx 2 +3xj ; dy 2x +3 = dx x 2 +3x (3) y =logjf(x)j ; dy dx = f 0 (x) f(x) 39

40 < 40 > (1) y =cos 1 x, x =cosy dy dx = 1 dy dx = 1 (cos y) 0 = 1 sin y = 1 p 1 cos 2 y = 1 p 1 x 2 (2) y =tan 1 x, x =tany dy dx = 1 dy dx = 1 (tan y) 0 = 1 1 cos 2 y = 1 1+tan 2 y = 1 1+x 2 40

(2000 )

(2000 ) (000) < > = = = (BC 67» BC 1) 3.14 10 (= ) 18 ( 00 ) ( ¼"½ '"½ &) ¼ 18 ¼ 0 ¼ =3:141596535897933846 ¼ 1 5cm ` ¼ = ` 5 = ` 10 () ` =10¼ (cm) (1) 3cm () r () () (1) r () r 1 4 (3) r, 60 ± 1 < > µ AB ` µ ±

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f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a 3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (

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